Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
The notion of adjunction of two variables is a natural generalization of both that of:
-enriched categories having powers and copowers.
Let , and be categories. An adjunction of two variables or two-variable adjunction
consists of bifunctors of this form
together with natural isomorphism of this form:
If is a two-variable adjunction, then so are
and
giving an action of the cyclic group of order 3. This can be made to look more symmetrical by regarding the original two-variable adjunction as a “two-variable left adjunction” ; see Cheng-Gurski-Riehl.
There is a straightforward generalization to an adjunction of variables, which involves categories and functors. Adjunctions of variables assemble into a 2-multicategory. They also have a corresponding notion of mates; see Cheng-Gurski-Riehl.
This 2-multicategory can also be promoted to a 2-polycategory; see Shulman.
Under the name “adjunction with a parameter” the concept appears in p. 102 of:
Under the name “THC-situation” the concept is discussed in:
The terminology adjunction of two variables is used in:
and:
Generalization to -variable adjunctions:
Eugenia Cheng, Nick Gurski, Emily Riehl, “Multivariable adjunctions and mates” [arXiv:1208.4520]
Rene Guitart, Trijunctions and triadic Galois connections. Cah. Topol. Géom. Différ. Catég. 54 (2013), no. 1, 13-27
Mike Shulman, The 2-Chu-Dialectica construction and the polycategory of multivariable adjunctions, arxiv:1806.06082, blog post
Last revised on December 16, 2023 at 17:13:21. See the history of this page for a list of all contributions to it.